Hostname: page-component-cd9895bd7-gxg78 Total loading time: 0 Render date: 2024-12-27T08:56:34.139Z Has data issue: false hasContentIssue false
  • English
  • Français

Gradient estimates for semilinear elliptic systems and other related results

Part of: Elliptic equations and systems

Published online by Cambridge University Press:  22 October 2015

Panayotis Smyrnelis
Panayotis Smyrnelis*
Affiliation:
Department of Mathematics, University of Athens, Panepistemiopolis, 15784 Athens, Greece ([email protected])
Get access Rights & Permissions [Opens in a new window]

Abstract

A periodic connection is constructed for a double well potential defined in the plane. This solution violates Modica's estimate as well as the corresponding Liouville theorem for general phase transition potentials. Gradient estimates are also established for several kinds of elliptic systems. They allow us to prove the Liouville theorem in some particular cases. Finally, we give an alternative form of the stress–energy tensor for solutions defined in planar domains. As an application, we deduce a (strong) monotonicity formula.

Keywords

Modica's estimateLiouville theoremgradient estimatesentire solutions to elliptic systemsstress–energy tensormonotonicity formula

MSC classification

Secondary: 35J47: Second-order elliptic systems
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 145 , Issue 6 , December 2015 , pp. 1313 - 1330
Copyright
Copyright © Royal Society of Edinburgh 2015 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)